| Model . | Score level . | Score variance . | Description . | |
|---|---|---|---|---|
| 1 | Restricted linear | hj + sk + e|${ijk}$| | |$\sigma ^2_h + \sigma ^2_s + \sigma ^2_e$| | Independent households & schools |
| 2 | Household upper bound | |$h_j+\gamma \bar{h}_{jk} + \nu _{k} + e_{ijk}$| | |$(1+\gamma )^2\sigma ^2_h + \sigma ^2_{\nu } + \sigma ^2_e$| | Household effects partly absorb school effects |
| 3 | School upper bound | |$s_k+\theta \bar{s}_{kj} + \omega _{j} + e_{ijk}$| | |$(1+\theta )^2\sigma ^2_s + \sigma ^2_{\omega } + \sigma ^2_e$| | School effects partly absorb household effects |
| 4 | Unrestricted linear | hj + sk + e|${ijk}$| | |$\sigma ^2_h + \sigma ^2_s + 2\Sigma _{hs} + \sigma ^2_e$| | Correlated household & school factors |
| Model . | Score level . | Score variance . | Description . | |
|---|---|---|---|---|
| 1 | Restricted linear | hj + sk + e|${ijk}$| | |$\sigma ^2_h + \sigma ^2_s + \sigma ^2_e$| | Independent households & schools |
| 2 | Household upper bound | |$h_j+\gamma \bar{h}_{jk} + \nu _{k} + e_{ijk}$| | |$(1+\gamma )^2\sigma ^2_h + \sigma ^2_{\nu } + \sigma ^2_e$| | Household effects partly absorb school effects |
| 3 | School upper bound | |$s_k+\theta \bar{s}_{kj} + \omega _{j} + e_{ijk}$| | |$(1+\theta )^2\sigma ^2_s + \sigma ^2_{\omega } + \sigma ^2_e$| | School effects partly absorb household effects |
| 4 | Unrestricted linear | hj + sk + e|${ijk}$| | |$\sigma ^2_h + \sigma ^2_s + 2\Sigma _{hs} + \sigma ^2_e$| | Correlated household & school factors |
Source: Authors’ elaboration.
Note: Variances in rows 3 and 4 assume |$\sigma _{h}^{2} \equiv {\rm{Var}}{(\textit h_{j})}\approx {{\rm Var}}{(\bar{h}_{jk})}$| and |${\rm Var}{(\textit s_{k})}\approx {\rm Var}{(\bar{s}_{\textit{kj}})}$|.
| Model . | Score level . | Score variance . | Description . | |
|---|---|---|---|---|
| 1 | Restricted linear | hj + sk + e|${ijk}$| | |$\sigma ^2_h + \sigma ^2_s + \sigma ^2_e$| | Independent households & schools |
| 2 | Household upper bound | |$h_j+\gamma \bar{h}_{jk} + \nu _{k} + e_{ijk}$| | |$(1+\gamma )^2\sigma ^2_h + \sigma ^2_{\nu } + \sigma ^2_e$| | Household effects partly absorb school effects |
| 3 | School upper bound | |$s_k+\theta \bar{s}_{kj} + \omega _{j} + e_{ijk}$| | |$(1+\theta )^2\sigma ^2_s + \sigma ^2_{\omega } + \sigma ^2_e$| | School effects partly absorb household effects |
| 4 | Unrestricted linear | hj + sk + e|${ijk}$| | |$\sigma ^2_h + \sigma ^2_s + 2\Sigma _{hs} + \sigma ^2_e$| | Correlated household & school factors |
| Model . | Score level . | Score variance . | Description . | |
|---|---|---|---|---|
| 1 | Restricted linear | hj + sk + e|${ijk}$| | |$\sigma ^2_h + \sigma ^2_s + \sigma ^2_e$| | Independent households & schools |
| 2 | Household upper bound | |$h_j+\gamma \bar{h}_{jk} + \nu _{k} + e_{ijk}$| | |$(1+\gamma )^2\sigma ^2_h + \sigma ^2_{\nu } + \sigma ^2_e$| | Household effects partly absorb school effects |
| 3 | School upper bound | |$s_k+\theta \bar{s}_{kj} + \omega _{j} + e_{ijk}$| | |$(1+\theta )^2\sigma ^2_s + \sigma ^2_{\omega } + \sigma ^2_e$| | School effects partly absorb household effects |
| 4 | Unrestricted linear | hj + sk + e|${ijk}$| | |$\sigma ^2_h + \sigma ^2_s + 2\Sigma _{hs} + \sigma ^2_e$| | Correlated household & school factors |
Source: Authors’ elaboration.
Note: Variances in rows 3 and 4 assume |$\sigma _{h}^{2} \equiv {\rm{Var}}{(\textit h_{j})}\approx {{\rm Var}}{(\bar{h}_{jk})}$| and |${\rm Var}{(\textit s_{k})}\approx {\rm Var}{(\bar{s}_{\textit{kj}})}$|.
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